For designing complex devices, it is valuable to be able to model systems of coupled resonators without the need for expensive numerical simulations (e.g. with commercial finite element packages). For this reason, there has been significant mathematical interest in developing concise models for coupled resonator systems. A prominent field in this direction is multiple scattering theory [1]}. These techniques are particularly effective for modelling either small (point) scatterers or systems whose geometry admits explicit representations (e.g. cylinders or spheres) [2]}, [3]}. To describe resonators with general, possibly complex, shapes, integral methods can be used [4]}. On top of this, asymptotic techniques have helped provide concise characterisations of complex problems. Homogenisation can be used characterise the effective properties of materials with periodic [5]}, [6]}, quasi-periodic [7]} or random [8]} micro-structures. Local properties can also be deduced through asymptotic approaches. For example, asymptotic expansions can be computed when resonators are very small or have highly contrasting material parameters [9]}, [10]}.
